Finding arc length of a parametric curve

Formula for arc length of a parametric curve

The arc length of a parametric curve over the interval ???\alpha\le{t}\le\beta??? is given by

???L=\int^\beta_\alpha\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\ dt???

where ???\alpha??? and ???\beta??? are the limits of the interval

where ???dx/dt??? is the derivative of ???x(t)???

where ???dy/dt??? is the derivative of ???y(t)???

Example

Find the length of the parametric curve over ???0\le{t}\le2\pi???.

???x=5\sin{t}???

???y=5\cos{t}???

We need to find the derivatives of the parametric equations.

???x=5\sin{t}???

???\frac{dx}{dt}=5\cos{t}???

and

???y=5\cos{t}???

???\frac{dy}{dt}=-5\sin{t}???

Since we were given the limits of integration in the problem, we’re ready to plug everything into the arc length formula.

???L=\int^{2\pi}_0\sqrt{(5\cos{t})^2+(-5\sin{t})^2}\ dt???

???L=\int^{2\pi}_0\sqrt{25\cos^2{t}+25\sin^2{t}}\ dt???

???L=\int^{2\pi}_0\sqrt{25(\cos^2{t}+\sin^2{t})}\ dt???

Since ???\cos^2{t}+\sin^2{t}=1???, we get

???L=\int^{2\pi}_0\sqrt{25(1)}\ dt???

???L=\int^{2\pi}_05\ dt???

???L=5t\big|^{2\pi}_0???

???L=5(2\pi)-5(0)???

???L=10\pi???